Turbulence Fingerprint on Collective Oscillations of Supernova Neutrinos

We bring to light a novel mechanism through which turbulent matter density fluctuations can induce collective neutrino flavor conversions in core-collapse supernovae, i.e., the leakage of flavor instabilities between different Fourier modes. The leakage mechanism leaves its notable fingerprint on the flavor stability of a dense neutrino gas by coupling flavor conversion modes on different scales which in turn, makes the flavor instabilities almost ubiquitous in the Fourier space. The most remarkable consequence of this effect is in that it allows for the presence of significant flavor conversions in the deepest supernova regions even in the absence of the so-called fast modes. This is yet another crucial impact of turbulence on the physics of core-collapse supernovae which can profoundly change our understanding of neutrino flavor conversions in the supernova environment.

can also experience the so-called fast flavor conversions on scales much shorter than those of traditional (slow) modes . The fast scales are determined by the neutrino number density, n ν , and can be as short as a few cm in the deepest SN zones, as opposed to the ones of slow modes which are determined by the vacuum frequency, ω = ∆m 2 /2E, and occur on scales of ∼ a few km (for ∆m 2 atm and E = 10 MeV neutrinos). Besides their phenomenological importance, perhaps the most remarkable physical consequence of fast modes is in that they can lead to the occurrence of collective neutrino oscillations in the deepest regions of the SN core. This is because they occur on short enough scales in such a way that the unstable modes can experience significant flavor conversions before the physical conditions vary significantly. In spite of their importance, fast modes do not seem to be a generic feature of CCSNe and even if they exist, they are thought to be present only in a finite region of the SN core [51][52][53][54][55][56]. Additionally, fast modes may also be less likely to occur in non-exploding SN models [57].
Turbulence plays a crucial role in CCSNe [58][59][60][61][62]. The impact of turbulent density fluctuations on neutrino oscillations has been extensively studied in 1D models [63][64][65][66][67][68][69][70][71][72][73], where it can induce flavor conversions through parametric resonances. Here, we demonstrate that the presence of turbulence in CCSNe can also induce collective neutrino flavor conversion modes via an entirely different mechanism, i.e., the leakage of flavor instabilities between different Fourier modes. This novel effect can significantly influence neutrino flavor evolution in the SN environment and in particular, it can lead to the presence of traditional (slow) collective neutrino oscillations in the deepest SN regions even in the absence of fast modes. What makes this novel effect more promising is in that it survives even for tiny turbulence amplitudes.
Linear Stability Analysis.-We start by deriving the equation of neutrino flavor evolution in the linear regime, in the two-flavor scenario where the flavor content of a neutrino can be described as where f ν 's are the neutrino initial occupation numbers arXiv:2007.13655v1 [astro-ph.HE] 27 Jul 2020 and, S and s carry information on neutrino flavor coherence and conversion, respectively. In the absence of collisions, the flavor evolution of the neutrino gas can be described by the Liouville-von Neumann equation (c = = 1) [74][75][76][77][78] i where v is the neutrino velocity and λ = √ 2G F n e is the matter contribution to the neutrino Hamiltonian [79,80]. Here, the energies and occupation numbers are taken to be positive for neutrinos and negative for antineutrinos, σ 3 is the third Pauli matrix and is the contribution from the neutrino-neutrino forward scattering [81][82][83].
We are here interested in the flavor stability analysis of neutrinos in the linear regime where the flavor conversion is still insignificant and one has s 1 and |S| 1. By only keeping terms of O(|S|) in Eq. (2), one reaches [84,85] i Eq. (4) provides a linear set of equations for which one can try collective solutions of the form S E,v = Q Ω,k E,v e −iΩt+ik·x where Ω and k satisfy the dispersion relation (DR) equation corresponding to Eq. (4). In a homogenous medium, this leads to Note that different Fourier modes are decoupled which means that k is just a parameter here and one only needs to find the functional form of Q Ω,k E,v in the E − v space for a solution of DR equation.
Turbulent Matter Fluctuations.-It simply follows from Eq. (6) that in a homogenous medium where matter is constant, the matter potential λ can be absorbed in the real part of Ω and therefore, does not affect the stability condition of the dense neutrino gas. However, if the matter is not constant and spatial density fluctuations are present, Eq. (6) changes to where λ k is the Fourier component of the matter potential. Note that, most remarkably, different Fourier modes are now coupled through the turbulence-induced convolution term and simple plane waves are not anymore eigenvectors of Eq. (7). This implies that in order to solve Eq. (7), one should also consider the distribution of Q Ω,k E,v in the Fourier space because k is not a parameter anymore and eigenvectors of Eq. (7) can now have contributions from a range of k's.
In the following, we assume a Kolmogorov-like spectrum for turbulence where the matter density features power-law fluctuations [86]. Such fluctuations can be present on a range of scales between the cutoff scale, λ cut , which is the longest wavelength of the turbulence and the dissipation scale, λ diss , which is the shortest wavelength of the turbulence below which the turbulent energy gets efficiently dissipated by viscosity. The cutoff scale is specified by the biggest relevant scale in the problem, which can be determined by the shock radius, i.e. λ cut ∼ 2R s . On the other hand, the dissipation scale can be estimated by considering the typical values of the matter density, the temperature and the fluid velocity in the SN zones above the proto-neutron star (PNS). Following Ref. [58], it turns out that λ diss 10 −10 km (k diss 10 10 km −1 ) in the deepest regions of the SN core.
To be specific, we take the turbulent matter fluctuations to have the form where λ 0 is the average matter potential (zeroth mode), η k is a random phase and C provides a measure of the overall amplitude of the turbulence. In addition, ξ k is assumed to have a Kolmogorov distribution with k cut = 2π/λ cut which is fixed to be k cut = 0.01 km −1 in our calculations. We also set α = 5/3 though our results do not depend qualitatively on the value of α for reasonable choices. With these choices, one has λ k ∼ Cλ 0 k/k cut −α/2 .
Two-Beam Model.-We study neutrino flavor instabilities in a stationary 2D two-beam, monochromatic neutrino gas studied first in Ref. [21] (this stationary model is chosen for illustrative purposes, otherwise see Supplemental Material for the turbulence effect on temporal instabilities). Such a model can be used to describe the SN geometry at some distance from the SN core [25] where a periodic boundary condition is imposed in the transverse plane (along the x-axis in our model) and we study the evolution of the neutrino gas along the z-axis which can also be interpreted as being the radial direction in spherical coordinate. The mono-energetic ν e and ν e beams (ω = ±1) are assumed to be emitted with (corresponding to an opening angle of 2π/3 between the two beams) and the ratio between the number densities is fixed to be nν e /n νe = 0.7. We solve Eq. (7) for our stationary model (Ω = 0) to find unstable modes in the z-direction (where the imaginary part of k z is positive) and the E − v distributions of the corresponding eigenvectors as a function of the Fourier mode in the x-direction, k (hereafter we drop the subscripts x and superscripts Ω). In addition, for the small turbulence amplitudes considered here one can safely ignore the turbulence in the z (longitudinal) direction since it is completely suppressed by the other terms in the equation of motion. This implies that Fourier modes are only coupled in the x (transverse) direction. Note that a similar suppression does not exist for the turbulence in x-direction since there is no other term being able to compete with the turbulence (coupling) effect.
To illustrate how turbulent matter density fluctuations impact the stability of a dense neutrino medium, in Fig. 1 we indicate the overall shape of the eigenvectors of Eq. (7), defined as corresponding to the unstable mode with the maximum growth rate, where the eigenvectors are normalized to have unit length. In the left panel, we first consider a neutrino gas with a relatively low neutrino number density, µ = √ 2G F n νe = 50 km −1 . For such values of µ, only very low Fourier modes are unstable in a homogenous neutrino gas. However, the instability structure changes dramatically in a turbulent medium. As expected, there is a dominant peak with |Q k | 1 at k 0. However, due to the presence of turbulent matter fluctuations, one can clearly see that the instability can now leak to much higher Fourier modes (and make them unstable) which are otherwise completely stable in the homogenous case. It is of utmost importance to note that in spite of its small amplitude (for the tiny turbulence amplitudes used here), the leakage of instabilities can entirely change the stability condition of the neutrino gas. This simply comes from the fact that as far as the flavor stability is concerned, the amplitude (of the unstable modes) is not relevant since any unstable mode can growth exponentially with a growth rate of ∼ 10 km −1 (for slow modes). Thus, even unstable modes with amplitude |Q k | ∼ 10 −9 can get activated within only ∼ 2 km. This implies that, surprisingly, even a tiny amount of turbulence in matter would be enough to have a notable influence and make the whole range of (relevant) Fourier modes unstable with reasonable initial amplitudes. This is very different from the turbulence-induced parametric resonances where turbulent matter fluctuations can only generate a noticeable effect when the turbulence amplitude is considerably large [64]. To the best of our knowledge, the flavor instability leakage is the only physical effect in CCSNe which is sensitive to such tiny turbulence amplitudes. Indeed, the turbulence effect behaves here like a switch-on effect. Thus, one might be tempted to interpret the leakage phenomenon as an example of the effect of the background symmetry breaking in a dense neutrino gas. Note that in the absence of turbulence, |Q k | should be a δ-function in the Fourier space.
The turbulence-induced leakage amplitude is almost independent of µ and depends only on the density fluctuation amplitude (see Supplemental Material) By using this expression, one can easily make an estimate of the leakage amplitude for a given matter density and turbulence amplitude.
In the middle panel of Fig. 1, an example of the instability leakage for a high neutrino number density with µ = 10 4 km −1 is presented. For such a value of µ which is expected in the SN zones close to the surface of the PNS, only very large k's are unstable in the homogenous case. However, the instability leaks to small k's in the presence of turbulence. In particular, the leakage amplitude for a given turbulence amplitude is much larger in this case since the matter density is quite big in the vicinity of the PNS.
Although the form of the eigenvectors of Eq. (7) changes significantly in a turbulent medium, its eigenvalues do not change noticeably, at least for such small turbulence amplitudes tried here. This implies that this novel effect observed for constant µ's might be still superficial unless it can also leave its influence on the solutions of Eq. (4) for a realistic SN profile where µ is changing. But this is exactly where the power of the leakage mechanism is best manifested, as illustrated in the right panel of Fig. 1. Here to provide a flavor of this effect, we show the evolution of the k = 10 3 km −1 Fourier mode in a model in which the neutrino number density is varying with µ(z) = 10 4 exp(−0.3z) km −1 (note that µ changes very rapidly and goes from 10 4 to 10 km −1 within only ∼ 20 km). As can be clearly seen, the final amplitude of the Fourier modes (at the point they become dominant) in the presence of turbulence can be larger than those of the homogenous gas by many orders of magnitude. This is due to the fact that all the relevant Fourier modes can always grow exponentially in a turbulent medium in contrast to the homogenous gas in which each Fourier mode has a certain range of instability. This behavior is completely compatible with/predictable from what already observed in the left and middle panels of Fig. 1 and shows that the assessment based on the shape of the eigenvectors of Eq. (7) can be very useful in providing a sufficient insight on how Fourier modes grow. Note that the exact turbulence-induced enhancement in the amplitude of a Fourier mode depends on the duration on which the turbulence influences its evolution, which can be much longer for realistic SN profiles [87].
The evolution of the neutrino gas here is adiabatic to a very good degree in the sense that the scales on which the eigenvectors of Eq. (7) grow (exponentially) are much shorter than those of variations in µ (or in the shape of the eigenvectors themselves), i.e., κ −1 µ/(dµ/dr). One can then better understand the behavior observed in the right panel of Fig. 1 in an analytical way, assuming a perfect adiabaticity. In the perfect adiabatic limit, the solution of Eq. (4) at each step z = z 0 + ∆z can be obtained analytically from the one at z = z 0 by S(z 0 + ∆z) = i c i Ψ i e i(kz)i∆z where Ψ i and (k z ) i are the eigenvectors (which form a complete basis) and eigenvalues of the Hamiltonian of Eq. (7) at z = z 0 and c i 's are the expansion coefficients of S(z 0 ) = i c i Ψ i .
Such an analytical adiabatic solution (red curve) shows a very good agreement with the numerical solution of Eq. (4). One should note that the key point here is that the eigenvectors of Eq. (7) at two different steps are not exactly linearly independent of each other. In other words, each Ψ new i has contributions from all Ψ old j 's, i.e., Ψ new i = j c ij Ψ old j with c ij being rouphly the turbulence amplitude. This means that any unstable mode grows from an enhanced initial value (occurred during the growth of the modes which were previously unstable) which in turn ensures that all the Fourier modes always grow exponentially during the propagation of neutrinos. This is entirely in contrast to the homogenous case where the new unstable modes at each point are totally linearly independent of the old ones at a previous point and therefore, any exponential growth is present only within a certain period.
Discussion and Conclusions.-The turbulenceinduced leakage of flavor instabilities implies that the notion of µ − k instability band (see, e.g., Fig. 3 in Ref. [22]) developed in a homogenous neutrino gas is not very useful in a turbulent medium where what distinguishes different Fourier modes is actually only their initial amplitudes, |Q k |.
One can immediately observe that the leakage mechanism can well address one of the biggest issues with slow instabilities in the deepest SN zones. In particular, it dismisses the necessity of the occurrence of fast modes in order to observe significant flavor conversions near the PNS. To demonstrate this idea, in Fig. 2 we show the instability footprints of two representative Fourier modes as a function of the distance from the SN core and the turbulence amplitude, C, during the accretion phase of a CCSN 1 . Here we take a matter/neutrino density profile approximately similar to that of Ref. [25] in which µ(r) = µ R (R/r) 4 and λ(r) = λ R (R/r) 3 , where µ R and λ R are the neutrino and matter densities on the surface of the neutrinosphere, R, respectively, for which we have used R = 15 km, µ R = 2 × 10 5 km −1 and λ R = 6 × 10 7 km −1 (corresponding to a matter density of ρ 3 × 10 11 g cm −3 ). For very small turbulence amplitudes, the instability zones can be extremely narrow specially at small radii which prevents any significant flavor conversions therein. However, as C increases, the Fourier modes become unstable at all radii which means that they can grow many orders of magnitude (as in the right panel of Fig. 1) and easily enter the nonlinear regime. Hence, no matter whether fast modes exist or not, collective neutrino oscillations can occur within just a few 1 Here we define the unstable region for a given Fourier mode by requiring |Q k | > 10 −13 for the eigenvector of the mode with maximum growth rate. This is to ensure that the activation scale of a given mode is shorter than the variation scales of µ. Otherwise, such boundaries for the unstable regions are absolutely artificial. km above the PNS. In addition, unlike fast modes which can only exist in small SN regions and are less likely to occur in non-exploding models, turbulence-induced flavor conversion modes are ubiquitous and generic. This could have an important impact on the SN dynamics and the nucleosynthesis of heavy elements in CCSNe.
Apart from the crucial impact of turbulence on the flavor stability of a dense neutrino gas, its presence is also important in providing initial seeds for the unstable modes. Specifically, the turbulence term in Eq. 4 transfers the initial seed from k 0 to k 0 ± k on scales ∼ λ −1 k , or more accurately, ∼ max{λ −1 k k/k } where the maximum is taken over all turbulence modes.
Apart from its impact on slow modes which was discussed here, turbulence can also affect fast neutrino flavor conversion modes (see Supplemental Material). However, although a similar leakage can occur therein, the leakage mechanism does not seem to change DR of fast modes.
In the above discussions, we have only considered the effects of spatial density fluctuations on the spatial instabilities. However, a similar effect should also be expected for temporal instabilities as shown in Supplemental Material. Indeed, the leakage effect has nothing to do with the eigenvalues of DR equation and the nature of instabilities and, it only arises due to the presence of coupling among different eigenvectors. Additionally as discussed in Supplemental Material, such an effect even exists for stable solutions (real eigenvalues of DR equation). Similarly, temporal fluctuations of the matter density can also couple different temporal frequencies. Although extremely rapid temporal density variations are necessary to observe any noticeable effect, it could be still plausible considering the small required turbulence amplitudes. Moreover, while we have only considered the effects of the turbulence on flavor instability in CCSNe, similar effect can be expected in any dense neutrino environment where matter density fluctuations are present, such as neutron star merger remnant accretion disks.
Our study is meant only to serve as an introduction to this novel issue and further research is necessary to provide a better understanding of its physical implications. This is yet another time that the rich physics of neutrino flavor evolution in dense neutrino media surprises us.
Acknowledgments.-I am deeply indebted to Georg Raffelt and Huaiyu Duan for their comments on the manuscript, their kind encouragements and many insightful discussions during the development of this work. I am also grateful to Cristina Volpe, Ernazar Abdikamalov and Shashank Shalgar for useful conversations/communications and to Hans-Thomas Janka and Francesco Capozzi for reading the manuscript and their comments.
I acknowledge partial support by the Deutsche Forschungsgemeinschaft (DFG) through Grant No. SFB 1258 (Collaborative Research Center Neutrinos, Dark Matter, Messengers).

SUPPLEMENTAL MATERIAL I. THE LEAKAGE AMPLITUDE
We here develop a hand-waving understanding of the turbulence-induced leakage amplitude. We take the structure of the equation of neutrino flavor evolution in the linear regime and attempt to understand how the presence of a coupling between different Fourier modes changes the structure of the eigenvectors. For this purpose, we consider the following set of equations which resembles the evolution of the flavor coherence terms of two Fourier modes which are coupled where k = k 2 − k 1 . It can be easily shown that the eigenvectors of this set of linear equations have the form |Q| ∝ (1, λ k /k) and (λ k /k, 1) for λ k /k 1 (which is always the case for the Fourier modes of interest). This confirms that in the presence of a coupling term, any eigenvector will have a dominant component at a given Fourier mode and subdominant contributions from other modes with amplitude ∼ λ k /k.

II. TEMPORAL INSTABILITIES
We have studied the leakage mechanism in a stationary dense neutrino gas. Here, we demonstrate that a very similar effect arises in a time-dependent model where turbulence can impact the temporal instabilities. This should be of course expected from the nature of the leakage mechanism.
We here consider a time-dependent two-beam, monochromatic neutrino gas with one spatial dimension. Our model is the same as the one proposed in Ref. [40] but we only consider two (zenith) angle beams with emission angles ϑ 1 = π/6 and ϑ 2 = π/3 with respect to the z-axis. We here assume that the neutrino gas possesses a perfect axial symmetry about the z-axis.
The results obtained in this model are presented in Fig. 3 where we have as an example shown the overall shape of |Q k | (here k is the Fourier mode in z-direction) for µ = 50, corresponding to the unstable temporal mode with the maximum growth rate. As can be obviously seen, the temporal instabilities are similarly affected by the leakage mechanism.

III. FAST MODES
Turbulent matter fluctuations can also influence fast modes in a similar way to slow modes. This is clearly illustrated in Fig. 4 where two arbitrary stable and unstable modes are shown for a neutrino gas in the presence of fast modes. Here we have considered a 2D stationary  7) for the time-dependent neutrino gas described here, corresponding to the unstable temporal mode with the maximum growth rate for µ = 50 km −1 . Here to relate matter to neutrino number density we have used Eq. (12).
neutrino gas with only one neutrino and one antineutrino beams with vν e = (+u, 0, v z ) and v νe = (−u, 0, v z ) (v z = 1/2) in such a way that fast modes can exist (see the model studied in Ref. [33]). The blue and red bands indicate the regions where the real branches and the gap (where complex branches exist) are located, respectively, in a homogenous neutrino gas. In the presence of turbulence, both the real and complex solutions can leak to the other zone. Note that the leakage phenomenon is not unique to the unstable modes (complex branches) and stable modes also leak to the unstable region (with |Q k | ∼ λ k /k). Note, however, that considering the locality of fast modes, as long as the eigenvalues of Eq. (7) are not modified by the turbulence one can always define a new set of basis so that the DR remains unchanged. Overall shape of two arbitrary real and complex solutions of Eq. (7), in the presence of fast modes in a turbulent medium, with C = 10 −8 . The blue and red areas show the regions where the real branches and the gap (the complex branches) exist in a homogenous neutrino gas, respectively.